2012/09/20 by Finnur Lárusson, Finnur Larusson, Tyson Ritter +2
Mathematics · #32H02 (Primary) 32E10 #32H35 #32M17 #32Q28 #32Q40 (Secondary) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Holomorphic and Operator Theory #math.CV #msc:32E10 #msc:32H02 #msc:32H35 #msc:32M17 #msc:32Q28 #msc:32Q40
paper · pdf · doi:10.48550/arxiv.1209.4430
openalex publication_date 2012/09/20 · arxiv created 2012/10/24 · arxiv updated 2012/10/26 · openalex created_date 2022/09/26 · openalex updated_date 2026/07/28
Gromov, in his seminal 1989 paper on the Oka principle, proved that every continuous map from a Stein manifold into an elliptic manifold is homotopic to a holomorphic map. Previously we have shown that, given a continuous map X → \C×\C^* from a finitely connected planar domain X without isolated boundary points, a stronger Oka property holds, namely that the map is homotopic to a proper holomorphic embedding. Here we show that every continuous map from a finitely connected planar domain, possibly with punctures, into \C×\C^* is homotopic to a proper immersion that identifies at most countably many pairs of distinct points, and in most cases, only finitely many pairs. By examining situations in which the immersion is injective, we obtain a strong Oka property for embeddings of some classes of planar domains with isolated boundary points. It is not yet clear whether a strong Oka property for embeddings holds in general when the domain has isolated boundary points. We conclude with some observations on the existence of a null-homotopic proper holomorphic embedding \C^* → \C×\C^*.